Inverses

An inverse function is a function that undoes another function, much like addition undoes subtraction. Inverse functions use inverse operations to turn inputs into outputs and vice versa.

The function, f ^-1 (x), is defined as the inverse function of f(x) if it consistently reverses the process of f(x). That is, if f(x) turns a into b, then f ^-1(x) must turn b into a. f ^-1 (x) is not an exponent and is read "the inverse of function f." It is meant to be an inverse function, not a reciprocal.

There are a couple of ways to think about the inverse of a function. Inverses can be approached by looking at graphs or performing algebraic operations. In either case, it comes down to the basic notion that the inverse of a function reverses the x and y coordinates. In other words, for every ordered pair (x, y) in a function there will be an ordered pair (y, x) in the inverse function.

The graph of an inverse of a function is reflected over the line y = x. Reflecting over the line y = x produces the goal of reversing the x and y coordinates.

In the graph below, the original function is reflected over the line y =x and gives us the inverse function .

Example 1 If a function answers the question: "I have this many spoons, how much do they weigh?" The inverse answers the question: "My spoons weigh this much, how many do I have?"

Example 2 If a function answers the question: "How many hours of music fit on 12 CDs?" The inverse answers the question: "How many CDs do you need for 3 hours of music?"

Example 3 Alice makes $100/day. Use the functionm(t)=100t to answer the question "After 3 days, how much money has she made?"

How do you recognize an inverse function?

Let's look at the two functions below:

Mathematically, these are inverse functions because they reverse the inputs and outputs.

A common example of inverse functions is the Celsius-to-Fahrenheit conversion:

where C is the Celsius temperature and F is the Fahrenheit temperature. Input 100°C into the first equation and the output is 212°F. Input 212°F into the second equation the output is 100°C. Therefore they are inverses.

To find an inverse symbolically:

Example 1 Find the inverse of the function f(x) = 5x -2.

Find the inverse of

To learn how to find the inverse of a function, select the following link:

Writing Inverse Functions Using Inverse Operations

The inverse of a function differs from the function in that all the x-coordinates and y-coordinates have been switched. That is, if (4,6) is a point on the graph of the function, then (6,4) is a point on the graph of the inverse function. The graph of a function and its inverse are mirror images of each other. They are reflected about the line y = x.

To learn how to graph the inverse, select the following link:

Graphing Inverse Relations

To solidify your understanding of inverse functions, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Inverses Worksheet

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Inverses Answer Key

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